This invention relates generally to phase measurements of signals such as Magnetic Resonance Imaging (MRI) signals, and more particularly the invention relates to determining the absolute phase of a signal given its principal value.
Nuclear magnetic resonance (NMR) imaging, also called magnetic resonance imaging (MRI), is a nondestructive method for the analysis of materials and represents a new approach to medical imaging. It is completely non-invasive and does not involve ionizing radiation. In very general terms, nuclear magnetic moments are excited at specific spin precession frequencies which are proportional to the local magnetic field. The radio-frequency signals resulting from the precession of these spins are received using pickup coils. By manipulating the magnetic fields, an array of signals is provided representing different regions of the volume. These are combined to produce a volumetric image of the nuclear spin density of the body.
Briefly, a strong static magnetic field is employed to line up atoms whose nuclei have an odd number of protons and/or neutrons, that is, have spin angular momentum and a magnetic dipole moment. A second RF magnetic field, applied as a single pulse transverse to the first, is then used to pump energy into these nuclei, flipping them over, for example to 90.degree. or 180.degree.. After excitation the nuclei gradually return to alignment with the static field and give up the energy in the form of weak but detectable free induction decay (FID). These FID signals are used by a computer to produce images.
The excitation frequency, and the FID frequency, is defined by the Larmor relationship which states that the angular frequency .omega..sub.0, of the precession of the nuclei is the product of the magnetic field B.sub.0, and the so-called magnetogyric ratio, .gamma., a fundamental physical constant for each nuclear species: EQU .omega..sub.0 =B.sub.0 .multidot..gamma.
Accordingly, by superimposing a linear gradient field, B.sub.z =Z.multidot.G.sub.z, on the static uniform field, B.sub.0, which defined Z axis, for example, nuclei in a selected X-Y plane can be excited by proper choice of the frequency spectrum of the transverse excitation field applied along the X or Y axis. Similarly, a gradient field can be applied in the X-Y plane during detection of the FID signals to spatially localize the FID signals in the plane. The angle of nuclei spin flip in response to an RF pulse excitation is proportional to the integral of the pulse over time.
Phase unwrapping is the process of determining the absolute phase given its principal value. This seemingly simple task is made difficult by the presence of measurement noise and has been analyzed for many different applications, ranging from homomorphic signal processing to speckle imaging, and recently for magnetic resonance imaging (MRI).
The MR phase can be designed to be a measure of some physical quantity. Depending on the pulse sequence, the MR phase can represent the main B.sub.0 field inhomogeneity or can be proportional to the velocity of the moving spins. Phase unwrapping is a necessary tool for the three-point Dixon water and fat separation and can be used to increase the dynamic range of phase contrast MR velocity measurements. In this paper, we propose a phase unwrapping technique based on the solution of the Poisson equation and apply the algorithm to MR phase images.
Various disciplines require reliable phase unwrapping algorithms. Homomorphic signals processing, where a nonlinear processing is achieved by linear operations in the complex cepstral frequency domain, requires computation of the complex cepstrum. The cepstrum necessitates a continuous imaginary part of the logarithm (phase of the spectrum). Since the imaginary part of the logarithm is only available modulo 2.pi., the phase must be unwrapped prior to the Fourier transform. In a similar manner, inverse scattering algorithms based on the Rytov approximation require phase unwrapping to determine the imaginary part of the logarithm. Other applications of phase unwrapping can be found in interference pattern analysis and image restoration.
In speckle imaging, unwrapping the phase of the Fourier transform is required prior to averaging. In this application, the phase has been unwrapped simply by integrating the wrapped phase derivative. Although simple, this method is extremely sensitive to noise. Moreover, a noise spike introduced early on will remain throughout this phase tracking process.
In signal processing applications, it is typically desired to unwrap the phase of the Fourier transform, e.g., for the cepstral computation mentioned above. For this purpose, various 1-D phase unwrapping algorithms have been proposed: adaptive integration of the phase derivative, counting sign changes in a Strum sequence, using zero locations of a band-limited function, detecting a zero's crossing of the unit circle, and using a discrete optimal nonlinear filtering technique. Although extensions of these algorithms to 2-D are possible, for instance, by unwrapping a 2-D signal along a prescribed path, the algorithms are 1-D in nature. One-dimensional algorithms extended to 2-D only make use of the phase values of the neighbors that lie on a (1-D) path--other neighbors are ignored.
The limitations of 1-D phase unwrapping algorithms extended to 2-D are due to the fact that the measurement noise can cause the result to depend on the chosen path. This point has been discussed using the idea of branch cuts. It is known that branch cuts can be placed so that phase unwrapping along any path without crossing the branch cut will be path independent. Since there can be a large number of such branch cuts, the set of branch cuts with the shortest total length can be used to unwrap the phase. However, this shortest length solution is heuristic and cannot be valid for general applications.
The idea of computing the least squares phase given measurements of the principal values has been described. The least squares phase can be obtained by using an iterative algorithm. The use of a direct algorithm for computing the least square phase was proposed where the algorithm derived in the discrete domain can be viewed as a discretization of the Poisson equation.